Problem: If you roll a 12-sided die, what is the expected value you will roll?
The expected value of an event (like rolling a die) is the average of the values of each outcome. To get an accurate idea of what value to expect, we weight the value of each outcome according to its probability. In this case, there are 12 outcomes: the first outcome is rolling a 1, the second outcome is rolling a 2, and so on. The value of each of these outcomes is just the number you roll. So, the value of the first outcome is 1, and its probability is $\dfrac{1}{12}$ The value of the second outcome is 2, the value of the third outcome is 3, and so on. There are 12 outcomes altogether, and each of them occurs with probability $\dfrac{1}{12}$ So, if we average the values of each of these outcomes, we get the expected value we will roll, which is $\dfrac{1}{12}+\dfrac{2}{12}+\dfrac{3}{12}+\cdots+\dfrac{10}{12}+\dfrac{11}{12}+\dfrac{12}{12} = 6 \dfrac{1}{2}$.